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Mon. Not. R. Astron. Soc. 366, 101–114 (2006) doi:10.1111/j.1365-2966.2005.09782.x

Systematic errors in future weak-lensing surveys: requirementsand prospects for self-calibration

Dragan Huterer,1 Masahiro Takada,2 Gary Bernstein3 and Bhuvnesh Jain4

1Kavli Institute for Cosmological Physics and Astronomy and Astrophysics Department, University of Chicago, Chicago, IL 60637, USA2Astronomical Institute, Tohoku University, Sendai 980-8578, Japan3Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA4Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA

Accepted 2005 October 21. Received 2005 October 17; in original form 2005 June 20

ABSTRACTWe study the impact of systematic errors on planned weak-lensing surveys and compute therequirements on their contributions so that they are not a dominant source of the cosmologicalparameter error budget. The generic types of error we consider are multiplicative and additiveerrors in measurements of shear, as well as photometric redshift errors. In general, more power-ful surveys have stronger systematic requirements. For example, for a SuperNova/AccelerationProbe (SNAP)-type survey the multiplicative error in shear needs to be smaller than 1 per centof the mean shear in any given redshift bin, while the centroids of photometric redshift binsneed to be known to be better than 0.003. With about a factor of 2 degradation in cosmologicalparameter errors, future surveys can enter a self-calibration regime, where the mean systematicbiases are self-consistently determined from the survey and only higher order moments of thesystematics contribute. Interestingly, once the power-spectrum measurements are combinedwith the bispectrum, the self-calibration regime in the variation of the equation of state of darkenergy wa is attained with only a 20–30 per cent error degradation.

Key words: cosmological parameters – large-scale structure of Universe.

1 I N T RO D U C T I O N

There has been significant recent progress in the measurements ofweak gravitational lensing by large-scale structure. Only 5 yr afterthe first detections made by several groups (Bacon, Refregier &Ellis 2000; Kaiser, Wilson & Luppino 2000; van Waerbeke et al.2000; Wittman et al. 2000), weak lensing already imposes strongconstraints on the matter density relative to critical M and theamplitude of mass fluctuations σ 8 (Hoekstra, Yee & Gladders 2002;Jarvis et al. 2003; Heymans et al. 2004; Rhodes et al. 2004a; for areview see Refregier 2003), as well as the first interesting constraintson the equation of state of dark energy (Jarvis et al. 2005).

The main advantage of weak lensing is that it directly probesthe distribution of matter in the Universe. This makes weak lensinga powerful probe of cosmological parameters, including those de-scribing dark energy (Hu & Tegmark 1999; Huterer 2002; Heavens2003; Hu 2003a; Refregier 2003; Benabed & Van Waerbeke 2004;Ishak et al. 2004; Song & Knox 2004; Takada & Jain 2004; Takada& White 2004; Ishak 2005). The weak-lensing constraints are espe-cially effective when some redshift information is available for thesource galaxies; use of redshift tomography can improve the cos-mological constraints by factors of a few (Hu 1999). Furthermore,

E-mail: [email protected]

measurements of the weak-lensing bispectrum (BS) (Takada & Jain2004) and purely geometrical tests (Jain & Taylor 2003; Zhang, Hui& Stebbins 2003; Bernstein & Jain 2004; Hu & Jain 2004; Song& Knox 2004; Bernstein 2005) lead to significant improvements ofaccuracy in measuring the cosmological parameters. When thesemethods are combined, weak lensing by itself is expected to con-strain the equation of state of dark energy w to a few per cent, and toimpose interesting constraints on the time variation of w. Ongoingor planned surveys, such as the Canada–France–Hawaii TelescopeLegacy Survey1, the Dark Energy Survey2 (DES), PanSTARRS3 andVisible and Infrared Survey Telescope for Astronomy (VISTA)4 areexpected to significantly extend lensing measurements, while the ul-timate precision will be achieved with the SuperNova/AccelerationProbe5 (SNAP; Aldering et al. 2004) and the Large Synoptic SurveyTelescope6 (LSST).

Powerful future surveys will have very small statistical uncer-tainties due to the large sky coverage and huge number of galaxies,

1 http://www.cfht.hawaii.edu/Science/CFHLS.2 http://cosmology.astro.uiuc.edu/DES.3 http://pan-starrs.ifa.hawaii.edu.4 http://www.vista.ac.uk.5 http://snap.lbl.gov.6 http://www.lsst.org.

C© 2005 The Authors. Journal compilation C© 2005 RAS

102 D. Huterer et al.

and therefore an understanding of the systematic error budget willbe crucial. However, so far the rosy weak-lensing parameter accu-racy predictions that have appeared in literature have not allowedfor the presence of systematics [exceptions are Ishak et al. (2004)and Knox et al. (2005) who consider a shear calibration error, andBernstein (2005) who does the same for the cross-correlation cos-mography of weak lensing]. This is not surprising, as we are juststarting to understand and study the full budget of systematic errorspresent in weak-lensing measurements. Nevertheless, some recentwork has addressed various aspects of the systematics, both experi-mental and theoretical, and ways to correct for them. For example,Vale et al. (2004) estimated the effects of extinction on the extractedshear power spectrum (PS), while Hirata & Seljak (2003), Hoekstra(2004) and Jarvis & Jain (2004) considered the errors in measure-ments of shear. Several studies explored the effects of theoreticaluncertainties (Huterer et al. 2004; White 2004; Zhan & Knox 2005;Hagan, Ma & Kravtsov 2005) and ways to protect against their ef-fects (Huterer & White 2005; Huterer & Takada 2005). It has beenpointed out that second-order corrections in the shear predictions canbe important (Cooray & Hu 2002; Hamana et al. 2002; Schneider,Van Waerbeke & Mellier 2002; Dodelson & Zhang 2005; Dodelsonet al. 2005; White 2005).

Despite these efforts, we are at an early stage in our understand-ing of weak-lensing systematics. Realistic assessments of system-atic errors are likely to impact strategies for measuring the weak-lensing shear (Bernstein 2002; Bernstein & Jarvis 2002; Rhodeset al. 2004b; Ishak & Hirata 2005; Mandelbaum et al. 2005). Amajor effort to compare the different analysis techniques and theirassociated systematics is already underway (the Shear Testing Pro-gramme; Heymans et al. 2005). Eventually we would like to bringweak lensing to the same level as cosmic microwave background(CMB) anisotropies and type Ia supernovae, where the systematicerror budget is better understood and requirements for the control ofsystematic precisely outlined (e.g. Tegmark et al. 2000; Hu, Hedman& Zaldarriaga 2003; Kim et al. 2003; Linder & Miquel 2004).

The purpose of this paper is to introduce the framework for thediscussion of systematic errors in weak-lensing measurements andoutline requirements for several generic types of systematic error.The reason that we do not consider specific sources of error (e.g.temporal variations in the telescope optics or fluctuations in atmo-spheric seeing) is that there are many of them, they strongly dependon a particular survey considered, and they are often poorly knownbefore the survey has started collecting data. Instead we argue that,at this early stage of our understanding of weak-lensing systemat-ics, it is more practical and useful to consider three generic typesof error – multiplicative and additive errors in measurements ofshear, as well as redshift error. These generic errors are useful in-termediate quantities that link actual experimental sources of errorto their impact on cosmological parameter accuracy. In fact, real-istic systematic errors for any particular experiment can in generalbe converted to these three generic systematics. Given the speci-fications of a particular survey, one can then estimate how mucha given systematic degrades cosmological parameters. This can beused to optimize the design of the experiment to minimize the ef-fects of systematic errors on the accuracy of desired parameter. Forexample, accurate photometric redshift requirements will lead to re-quirements on the number of filters and their wavelength coverage.Similarly, the requirements on multiplicative and additive errors inshear will determine how accurate the sampling of the point spreadfunction needs to be.

The plan of this paper is as follows. In Section 2, we discuss thesurvey specifications and cosmological parameters in this study. In

Section 3, we describe the parametrization of systematic errors. InSections 4–6, we present the requirements on the systematic errorsfor the PS, while in Section 7 we study the requirements whenboth the PS and the BS are used. We combine the redshift andmultiplicative errors and discuss trends in Section 8 and concludein Section 9.

2 M E T H O D O L O G Y, C O S M O L O G I C A LPA R A M E T E R S A N D F I D U C I A L S U RV E Y S

We express the measured convergence PS as

Cκi j () = Pκ

i j () + δi j

σ 2γ

ni, (1)

where Pκi j () is the measured PS with systematics [see the next

section and equation (20) on how it is related to the no-systematicPS Pκ

i j ()], σ 2γ is the variance of each component of the galaxy shear

and ni is the average number of resolved galaxies in the ith redshiftbin per steradian. The convergence PS at a fixed multipole and forthe ith and jth redshift bins is given by

Pκi j () =

∫ ∞

0

dzWi (z) W j (z)

r (z)2 H (z)P

(

r (z), z

), (2)

where r(z) is the comoving angular diameter distance and H(z) isthe Hubble parameter. The weights Wi are given by

Wi (χ ) = 3

2M H 2

0 gi (χ ) (1 + z), (3)

where gi (χ ) = r (χ )∫ ∞

χdχsni (χs)r (χs −χ )/r (χs), χ is the comov-

ing radial distance and ni is the fraction of galaxies assigned to theith redshift bin. We employ the redshift distribution of galaxies ofthe form

n(z) ∝ z2 exp(−z/z0), (4)

where z0 is survey-dependent and specified below. The cosmologicalconstraints can then be computed from the Fisher matrix

Fi j =∑

(∂C

∂pi

)T

Cov−1 ∂C

∂p j, (5)

where C is the column vector of the observed power spectra andCov−1 is the inverse of the covariance matrix between the powerspectra whose elements are given by

Cov[Cκ

i j (′), Cκ

kl ()]

= δ′

(2 + 1) fsky

[Cκ

ik()Cκjl () + Cκ

il ()Cκjk()

]. (6)

Here is the bandwidth in multipole we use, and f sky is the frac-tional sky coverage of the survey.

In addition to any nuisance parameters describing the systemat-ics, we consider six or seven cosmological parameters and assumea flat universe throughout. The six standard parameters are energydensity and equation of state of dark energy DE and w, spectralindex n, matter and baryon physical densities Mh2 and Bh2, andthe amplitude of mass fluctuations σ 8. Note that w = constant pro-vides useful information about the sensitivity of an arbitrary w(z),since the best-measured mode of any w(z) is about as well measuredas w = constant and therefore is subject to similar degradations inthe presence of the systematics. It is this particular mode, being themost sensitive to generic systematics, that will drive the accuracy re-quirements – we explicitly illustrate this in Fig. 2. In addition to theconstant w case, we also consider a commonly used two-parameter

C© 2005 The Authors. Journal compilation C© 2005 RAS, MNRAS 366, 101–114

Systematic errors in future weak-lensing surveys 103

Table 1. Fiducial sky coverage, density of source galaxies, varianceof (each component of) shear of one galaxy, and peak of the sourcegalaxy redshift distribution for the three surveys considered.

DES SNAP LSST

Area (deg2) 5000 1000 15 000n (gal arcmin−2) 10 100 30

σ γ 0.16 0.22 0.22zpeak 0.5 1.0 0.7

description of dark energy w(z) = w0 + wa z/(1 + z) (Chevallier &Polarski 2001; Linder 2003) where wa becomes the seventh cosmo-logical parameter in the analysis. Throughout we consider lensingtomography with 7–10 equally spaced redshift bins (see below), andwe use the lensing power spectra on scales 50 3000. We holdthe total neutrino mass fixed at 0.1 eV; the results are somewhatdependent on the fiducial mass. We compute the linear PS usingthe fitting formulae of Eisenstein & Hu (1999). We generalize theformulae to w = −1 by replacing the Lambda cold dark matter(CDM) growth function of density perturbations with that for ageneral w(z); the latter is obtained by integrating the growth equa-tion directly (e.g. equation 1 in Cooray, Huterer & Baumann 2004).To complete the calculation of the full non-linear PS we use thefitting formulae of Smith et al. (2003).

The fiducial surveys, with parameters listed in Table 1, are: theDark Energy Survey; SNAP and LSST. Note that there is someambiguity in the definition of the number density of galaxies ng;it is the quantity σ 2

γ /ng that determines the shear measurementnoise level, where σγ is the intrinsic shape noise of each galaxy.Galaxies implicitly assumed for the DES shear measurements arethose with the largest angular sizes, and therefore they have corre-spondingly smaller intrinsic shape noise than the SNAP and LSSTgalaxies. The surveys are assumed to have the source galaxy dis-tribution of the form in equation (4) which peaks at zpeak = 2z0.For the fiducial SNAP and LSST surveys, we assume tomographywith 10 redshift bins equally spaced out to z = 3, as future pho-tometric redshift accuracy will enable relatively fine slicing in red-shift. For the DES, we assume a more modest seven redshift binsout to z = 2.1, reflecting the shallower reach of the DES whilekeeping the redshift bins equally wide (z = 0.3) as in the othertwo surveys. Finally, we do not use weak-lensing information be-yond = 3000 in order to avoid the effects of baryonic cooling(Huterer et al. 2004; White 2004; Zhan & Knox 2004) and non-Gaussianity (White & Hu 2000; Cooray & Hu 2001), both of whichcontribute more significantly at smaller scales. While there may beways to extend the useful -range to smaller scales without riskingbias in cosmological constraints (Huterer & White 2005), extendingthe measurements to max = 10 000 would improve the marginal-ized errors on cosmological parameters by only about 30 per cent.7

The parameter fiducial values and accuracies are summarized inTable 2. The fiducial values for the parameters not listed in Table 2are Mh2 = 0.147, Bh2 = 0.021, n = 1.0, and m ν = 0.1 eV.

It is well known that measurements of the angular PS of the CMB,such as those expected by the Planck experiment, can help weaklensing to constrain the cosmological parameters. In particular, themorphology of the peaks in the CMB angular PS contains useful

7 On the other hand, especially for the DES, the Gaussian covariance as-sumption may be somewhat optimistic for the range 1000 < < 3000 (e.g.White & Hu 2000).

information on the physical matter and baryon densities, while thelocations of the peaks help to constrain the dark energy parameters.However, we checked that, when the Planck prior added, all sys-tematic requirements become weaker (relative to those with weaklensing alone) since the Planck information is not degraded withsystematic errors even if weak-lensing information is. In order to beconservative, we decided not to add the Planck CMB information.Therefore, we consider the systematic requirements in weak-lensingsurveys alone, and note that the addition of complementary infor-mation from other surveys typically weakens these requirements.

3 PA R A M E T R I Z AT I O NO F T H E S Y S T E M AT I C S

As mentioned above, we consider three generic sources of error.We believe that the parametrizations we propose, especially for theredshift and multiplicative shear errors, are in general enough toaccount for the salient effects of any generic systematic. The addi-tive shear error is more model-dependent, and while we motivate aparametrization we believe is reasonable at this time, further theoret-ical and experimental work needs to be done to understand additiveerrors.

We parametrize the redshift, multiplicative shear and additiveshear errors as follows.

3.1 Redshift errors

Measurements of galaxy redshifts are necessary not only for theredshift tomography – which significantly improves the accuracy inmeasuring dark energy parameters – but also to obtain the fiducialdistribution of galaxies in redshift, n(z). Therefore, understandingand correcting for the redshift uncertainties is crucial, and compar-ison studies between currently used photometric methods, such asthat initiated by Cunha et al. (in preparation), are of the highestimportance.

It is important to emphasize that statistical errors in photometricredshifts do not contribute to the error budget if they are well char-acterized. In other words, if we know precisely the distribution ofphotometric redshift errors (i.e. all of its moments) at each redshift,we can use the measured photometric distribution, np(zp), to recoverthe original spectroscopic distribution, n(zs) very precisely. In prac-tice, we will not know the redshift error distribution with arbitraryprecision, rather we will typically have some prior knowledge ofthe mean bias and scatter at each redshift.

The quantity we consider in this paper is the uncalibrated redshiftbias, that is, the residual (after correcting for the estimated bias)offset between the true mean redshift and the inferred mean pho-tometric redshift at any given z. A more general description of theredshift error would include the scatter in the redshift error at each z.Such an analysis has recently been performed by Ma, Hu & Huterer(2005) who found that, even though the scatter is important as well,the mean bias in redshift is the dominant source of error.

Unlike Ma et al. (2005) who hold the overall distribution of galax-ies in redshift n(z) fixed and only allow variations in the tomographicbin subdivisions, we allow the redshift error to affect the overall n(z)as well. At this time, it is not clear how the photometric error willaffect the source galaxy distribution n(z) as it depends on how thesource galaxy distribution will be determined. We assume that thesource galaxy distribution n(z) is obtained from the same photomet-ric redshifts used to subdivide the galaxies into redshift bins. Analternative possibility is that information about the overall distri-bution of source galaxies is obtained from an independent source



Table 2. Cosmological parameter errors without the systematics (numbers preceding the slash in each box) and with sample systematics (numbers followingthe slash). For the systematics case we assumed redshift biases (described by Chebyshev polynomials) of 0.0005, together with the multiplicative errors of0.005. The errors are shown for the PS tomography only, and for the PS and the BS combined. Errors in other cosmological parameters (Mh2, Bh2, n) arenot shown.

Parameter Fiducial value DES (PS) DES (PS plus BS) SNAP (PS) SNAP (PS plus BS) LSST (PS) LSST (PS plus BS)

M 0.3 0.008/0.011 0.006/0.009 0.008/0.011 0.004/0.006 0.003/0.007 0.002/0.005w −1.0 0.092/0.120 0.035/0.060 0.058/0.081 0.027/0.036 0.029/0.053 0.010/0.023σ 8 0.9 0.010/0.012 0.006/0.008 0.008/0.011 0.005/0.008 0.004/0.006 0.003/0.004w0 −1.0 0.33/0.40 0.18/0.20 0.28/0.35 0.09/0.12 0.13/0.20 0.06/0.07wa 0 1.41/1.64 0.82/0.92 0.96/1.20 0.35/0.44 0.49/0.69 0.25/0.27

(say, another survey) while the internal photometric redshifts areonly used to subdivide the overall distribution into redshift bins.The two approaches, ours and that of Ma et al. (2005), are thereforecomplementary and both should be studied. It is reassuring that thetwo approaches give consistent results.

We consider two alternative parametrizations of the redshift er-ror: the centroids of redshift bins, and the Chebyshev polynomialexpansion of the mean bias in the zp–z s relation.

(i) Centroids of redshift bins. We first consider the centroid ofeach photometric redshift bin as a parameter. Any scatter in galaxyredshifts in a given tomographic bin will average out, to first orderleaving the effect of an overall bias in the centroid of this bin. (Recall,the part of the bias that is uncalibrated and has not been subtractedout is what we consider here.) As discussed by Huterer et al. (2004)in the context of number-count surveys, this approach captures thesalient effects of the redshift distribution uncertainty. Note, however,that the centroid shifts do not capture the ‘catastrophic’ errors wherea smaller fraction of redshifts are completely mis-estimated andreside in a separate island in the zp–zs plane.

We therefore have B new parameters, where B is the number ofredshift bins. To compute the Fisher derivatives for these parameters,we vary each centroid by some value dz, that is, we shift the wholebin by dz. As mentioned above, this procedure not only allows for thefact that the tomographic bin divisions are not perfectly measured,but it also deforms the overall distribution of galaxies n(z).

(ii) Expansion of the redshift bias in Chebyshev polynomials.While the required accuracy of redshift bin centroids provides use-ful information, it is sometimes difficult to compare it with directlyobservable quantities. In reality, an observer typically starts withmeasurements of photometric redshifts which are not equal to thetrue, spectroscopic ones: the quantity zp−z s may have a non-zerovalue – the bias – and also non-zero scatter around the biased value.Therefore, it is sometimes more useful to consider requirements onthe accuracy in the zp–z s relation.

Detailed forms of the bias and scatter are typically complicatedand depend on the photometric method used and how well we areable to mimic the actual observations and correct for the biases.We write the redshift bias as a sum of Ncheb smooth functions –Chebyshev polynomials centred at zmax/2 and extending from z = 0to zmax, where zmax (3.0 for SNAP and LSST, 2.1 for the DES) is theextent of the distribution of galaxies in redshift. (We briefly reviewthe Chebyshev polynomials in Appendix A.) The relation betweenthe photometric and true, spectroscopic redshifts is then

zp = zs +Ncheb∑i=1

gi Ti

(z∗

s

), (7)

where

z∗s ≡ zs − zmax/2

zmax/2, (8)

and gi are the coefficients that parametrize the bias. As with thecentroids of redshift bins, we do not model the scatter in the zp–z s

relation as one can show that the effect of uncalibrated bias is domi-nant (Ma et al. 2005). The effect of imperfect redshift measurementsis to shift the distribution of galaxies away from the true distributionn(zs) to a biased one np(zp). The biased distribution of galaxies thenpropagates to bias the cosmological parameters. The photometricdistribution can then simply be obtained from the true distributionas (e.g. Padmanabhan et al. 2005)

np(zp) =∫ ∞

0

n(z)(zp − z, z) dz, (9)

where (zp − z, z) is the probability that the galaxy at redshift z ismeasured to be at redshift zp. Since we are not modelling the scatterin the zp–z s relation, the probability is a delta function

(zp − zs, zs) = δ

(zp − zs −

Ncheb∑i=1

gi Ti

(z∗

s

)). (10)

Since we will include the gi as additional parameters, with fidu-cial values gi = 0, we will need to take derivatives with onenon-zero gi at a time and therefore we can assume (zp − z s,z s) = δ(zp − z s − giTi(z∗

s )) for a single i. Using the fact thatδ(F(x)) = ∑

1/|F ′(x0)| δ(x − x0) for a function F(z), where thesum runs over the roots of the equation F(x) = 0, and further usinga recursive formula for the derivative of the Chebyshev polynomial(Arfken & Weber 2000, Section 13.3), we get

np(zp) =∑a

n(za)∣∣1 + 2gi/zmax

[1 − (

z∗a

)2][ − i za Ti

(z∗

a

) + iTi−1

(z∗

a

)]∣∣ ,(11)

where the sum runs over all roots of the equation z + giTi(z∗) − zp =0, with (recall) z∗

a ≡ (za − zmax/2)/(zmax/2). This is the expressionfor the perturbed distribution of galaxies due to a single perturbationmode.8 We use it, together with the original distribution n(z), tocompute the perturbed and unperturbed convergence power spectraand thus take the derivative with respect to gi.

In Fig. 1, we show the photometric galaxy distributions np(zp)for the selected three Chebyshev modes (first, second and seventh)of perturbation to the distribution of galaxies n(z) correspondingto Fig. A1, together with the original unperturbed distribution. The

8 Note that, for gi 1, equation (11) simplifies to np(zp) = n[zp − giTi(z∗p)].



0 0.5 1 1.5 2 2.5 3z

p

0

0.5

1n p(z

p)

1

2

7

DES

0 0.5 1 1.5 2 2.5 3z

p

0

0.5

1

n p(zp)

1

27

SNAP

Figure 1. Select three modes (first, second and seventh) of perturbation to the distribution of galaxies n(z) for the DES (left-hand panel) and SNAP (right-handpanel), shown together with the original unperturbed distribution (dashed curve in each panel). These three modes correspond to the modes of perturbation inthe zp–z s relation shown in Fig. A1 in Appendix A. Note that the fiducial n(z) of SNAP is broader, making the resulting wiggles in np(zp) less pronounced andthe weak-lensing measurements therefore less susceptible to the redshift biases. Note also that the allowed perturbations to n(z) are not only located near thepeak of the distribution, but can also have significant wiggles near the tails of the distribution.

distributions are shown for the DES and SNAP. Since the incorrectassignment of photometric redshifts only redistributes the galaxies,we always impose the requirement

∫ ∞0

np(zp) dzp = 1 regardless ofthe perturbation. Note that fiducial n(z) of SNAP is broader, mak-ing the resulting wiggles in np(zp) less pronounced and helping theweak-lensing measurements be less susceptible to the redshift bi-ases. (As shown in Section 4.1, this effect is counteracted by thesmaller fiducial errors in the SNAP survey which lead to more sus-ceptibility to biases.)

3.2 Multiplicative errors

The multiplicative error in measuring shear can be generated by avariety of sources. For example, a circular point spread function(PSF) of finite size is convolved with the true image of the galaxyto produce the observed image, and in the process it introduces amultiplicative error. Let γ (zs,n) and γ (z s, n) be the estimated andtrue shear of a galaxy at some true (spectroscopic) redshift zs anddirection n. Then the general multiplicative factor fi(θ ) acts as

γ (zs,n) = γ (zs,n)[1 + fi (zs,n)], (12)

where fi is the multiplicative error in shear which is both direction-dependent and time-dependent. We can write the error as a sum of itsmean (average over all directions and redshifts in that tomographicbin) and a component with zero mean,

f (zs,i,n) = fi + ri (n), (13)

where 〈 f (z s,i, n)〉 = fi and 〈ri(n)〉 = 0 and the averages are takenover angle and over redshift within the ith redshift bin. Then we canwrite

〈γ (zs,i,n) γ (zs,j,n + dn)〉= 〈γ (zs,i,n) γ (zs,j,n + dn) (1 + f (zs,i,n)) (1 + f (zs, j ,n))〉 (14)

〈γ (zs,i ,n)γ (zs, j ,n + dn)〉(1 + fi + f j ), (15)

where we dropped terms of order 〈γ γ fi fj〉. Since the random com-ponent of the multiplicative error is uncorrelated with shear, all termsof the form 〈γ γ r〉 are zero. Finally, terms of the form 〈γ γ rr〉 aretaken to be small, thus requiring that 〈ri (n) r j (n + dn)〉 is smaller

than fi and fj (Guzik & Bernstein 2005). Therefore,

Pκi j () = Pκ

i j ()[1 + fi + f j ], (16)

where again we emphasize that fi is the irreducible part of the mul-tiplicative error in the ith redshift bin (i.e. its average over all direc-tions and the ith redshift slab). It is clear that multiplicative errorsare potentially dangerous, since they lead to an error that goes as fi

and not f 2i . Finally, note that shear calibration is likely to depend on

the size of the galaxy, so that the mean error fi is the error averagedover all galaxy sizes in bin i.

3.3 Additive errors

Additive error in shear is generated, for example, by the anisotropyof the PSF. We define the additive error via

γ (zs,n) = γ (zs,n) + γadd(zs,n). (17)

Let us assume that the additive error is uncorrelated with the trueshears so that the term 〈γ (n)γ add(n)〉 is zero. Furthermore, let theLegendre transform of the term 〈γ add(n)γ add(n)〉 be Pκ

add(). Thenwe can write

Pκi j () = Pκ

i j () + Pκadd(). (18)

We now have to specify the additive systematic power Pκadd ().

Consider some known sources of additive error: for example, theadditive error induced due to a non-circular PSF is roughly R ePSF

where ePSF is the ellipticity of the PSF and R ≡ (sPSF/sgal)2 is theratio of squares of the PSF and galaxy size (E. Sheldon privatecommunication); galaxies that are smaller therefore have a largeradditive error. Of course, the observer needs to correct the overalltrend in the error, leaving only a smaller residual, and the impactof this residual is what we are interested in. It should be clear fromthis discussion that the additive error in shear can be described by apart that depends on the average size of a galaxy at a given redshiftmultiplied by a random component that depends on the part of thesky observed. Therefore, we write the additive shear asγ add(z s,i,n)=bir(n) where bi is the characteristic additive shear amplitude in theith redshift bin and r(n)is a random fluctuation.

In multipole space, the two-point function can then be expressedas bibj times the angular part which only depends on |ni −n j |. Notethat the additive error for two galaxies at different redshifts (i.e.



when i = j) is not zero, although it may in principle be suppressedrelative to the additive error for the auto-power spectra (when i = j).Motivated by such considerations, we assume the additive system-atic in multipole space of the form

Pκadd,i j () = ρ bi b j

(

∗

)α

, (19)

where the correlation coefficient ρ describes correlations betweenbins. The coefficient ρ is always set to unity for i = j, and for i = jit is fixed to some fiducial value (not taken as a parameter in theFisher matrix). While the discussion above would imply that ρ = 1for all i and j, we allow for the possibility that the additive errors arenot perfectly correlated across redshift bins. In the extreme case ofuncorrelated additive errors in different z bins, the cross-power spec-tra are not affected, ρ = δ i j . We will see that the results are weaklydependent on the fiducial value of ρ unless if ρ = 1 identically.

For the multipole dependence of Pκadd we assume a power-law

form, and marginalize over the index α. We choose ∗ = 1000, nearthe ‘sweet spot’ of weak-lensing surveys, but note that this choice isarbitrary and made solely for our convenience since ∗ is degeneratewith the parameters bi. Our a priori expectation for the value of α

is uncertain, and we try several possibilities but find (as discussedlater) that the results are insensitive to the value of α. We thereforehave a total of B + 1 nuisance parameters for the additive error(B parameters bi, plus α).

In practice there is no particular reason why an additive systematicwould be expected to conform to a power law, so in the future weshould consider an additive systematic with more freedom in itsspatial behaviour. The current treatment is optimistic: fixing thesystematic to a power law gives it an identifying characteristic thatallows us to distinguish it from cosmological signals. Without sucha characteristic, the systematic could be much more damaging.

3.4 Putting the systematics together

With the systematic errors we consider, the resulting theoreticalconvergence PS is

Pκi j () ≡ Pκ

[, z(i)

s , z( j)s

]= Pκ

(, z(i)

s + δz(i)p , z( j)

s + δz( j)p

)[1 + fi + f j ] + ρ bi b j

(

∗

)α

.

(20)

The derivatives with respect to the multiplicative and additiveparameters are trivial since they enter as linear (or power-law) coef-ficients, while the derivatives with respect to the redshift parametersδz(k)

p (really, parameters gi defined in equation 7) need to be takennumerically.

4 R E S U LT S : R E D S H I F T S Y S T E M AT I C S

4.1 Centroids of redshift bins

We have a total of N + B parameters, and we marginalize over theB redshift bin centroids by giving them identical Gaussian priors.While the actual redshift accuracy may be better in some redshiftranges than in others, the required accuracy that we obtain in ourapproach will pertain to the redshift bins where observations aremost sensitive (i.e. at intermediate redshifts z ∼ 0.5–1).

Fig. 2 shows the degradation in M, σ 8 and w = constant, andalso w0 and wa, accuracies as a function of our prior knowledgeof the redshift bin centroids. Here in Fig. 2 and in the subsequent

two figures, the plots are shown separately for each of the three sur-veys (DES, SNAP and LSST). Since we are using the Fisher matrixformalism without external priors on cosmological parameters, thecosmological parameter degradations (as well as accuracies) clearlyremain unchanged for an arbitrary f sky if we also scale the priors onthe nuisance parameters by f −1/2

sky . In these and subsequent plots weshow the degradations for fiducial values of the sky coverage foreach survey f sky,fid, where f sky,fid = f 5000, f 1000 and f15 000 for DES,SNAP and LSST, respectively, while indicating that the requirementsscale as ( f sky/ f sky,fid)−1/2.

We generally find that the degradations in different cosmolog-ical parameters are comparable. To have less than ∼50 per centdegradation, for example, we need to control the redshift centroidbias to about 0.003( f sky/ f sky,fid)−1/2 for the DES and SNAP and toabout 0.0015( f sky/ f sky,fid)−1/2 for the LSST. The current statisticalaccuracy in individual galaxy redshifts is of the order of 0.02–0.05.Averaging the large number of galaxies in a given redshift bin, itmay be that we are already close to the aforementioned requirementon the centroid bias, but this will of course depend on the control ofsystematics in the photometric procedure.

As mentioned in the Introduction, the accurately determined com-bination of w0 and wa , F(w0, wa) ≈ w0 + 0.3wa , is degraded innearly the same way as constant w; this is illustrated by a thickdashed line in each panel of Fig. 2. In the rest of this paper we donot repeat plotting the curves corresponding to F(w0, wa) whichnearly overlap those corresponding to w = constant. Finally, weshow the degradations for w0 and wa separately. Since these twoparameters are determined to a substantially lower accuracy thanw = constant (see Table 2), it is not surprising that the degradationsin these two parameters are substantially smaller.

4.2 Chebyshev expansion in zp–zs

Fig. 3 shows the degradation in the M, σ 8 and w, and also w0

and wa, as we marginalize over a large number (Ncheb = 30) ofChebyshev coefficients gi each with the given prior. We checkedthat, when only N 5 modes are allowed to vary, all parameters gi

can be self-consistently solved from the survey – this is an exampleof self-calibration. However, with a larger number of Chebyshevmodes the self-calibration regime is lost. Since we are interestedin the most general form of the redshift bias, this is the regime wewould like to explore. We have also checked that, as we let thenumber of coefficients increase further, the errors in cosmologicalparameters do not increase indefinitely, as the rapid fluctuations inthe zp − z s are not degenerate with cosmology. In fact we foundthat the degradations have asymptoted to their final values once weinclude 20–30 Chebyshev polynomials; therefore, fluctuations inredshift on scales smaller than z ∼ 0.1 are not degenerate withcosmological parameters. The choice of N cheb = 30 coefficients istherefore conservative.

Fig. 3 shows that the photometric redshift bias in each Chebyshevmode, for the DES and SNAP, should be controlled to about0.001( f sky/ f sky,fid)−1/2 or less. For LSST, the requirement is moresevere and we could tolerate biases no larger than about 0.0005( f sky/ f sky,fid)−1/2. These are fairly stringent requirements, in roughagreement with results found by Ma et al. (2005).

It might seem a bit surprising that the SNAP and DES require-ments are comparable, given that the fiducial SNAP survey is morepowerful than the DES (see Table 2) and hence would need bettercontrol of the systematics. However, there is another effect at playhere: a narrow distribution of galaxies in redshift, such as that from



0 0.002 0.004 0.006 0.008 0.01

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wa

Figure 2. Degradation in the cosmological parameter accuracies as a function of our prior knowledge of δz ≡ zp − z s. We assume equal Gaussian priors toeach redshift bin centroid, shown on the x-axis. For example, to have less than ∼50 per cent degradation in M, σ 8 or w, we need to control the redshift bias toabout 0.003 ( f sky/ f sky,fid)−1/2 or better for the DES and SNAP, and to about 0.0015 ( f sky/ f sky,fid)−1/2 or better for the LSST. For the varying equation-of-stateparametrization, the requirements for the best-measured combination of w0 and wa (F(w0, wa) ≈ w0 + 0.3wa) are identical to those for w = constant, whilethe requirements on w0 and wa individually are somewhat less stringent.

the DES, is more strongly subjected to fluctuations in zp − z s than awide distribution, such as that from SNAP; (see Fig. 1). Therefore,surveys with wide leverage in redshift have an advantage in beatingdown the effect of redshift error, just as in the case of cluster countsurveys (Huterer et al. 2004).

We now confirm the results by computing the bias in the cos-mological parameters due to the bias in one of the redshift biascoefficients, dgi. The bias in the cosmological parameter pα can becomputed as

δ pα = F−1αβ

∑

[Cκ

i () − Cκi ()

]Cov−1

[Cκ

i (), Cκj ()

] ∂C j ()κ

∂pβ

,

(21)

where the summations over β, i and j were implicitly assumedand the covariance of the cross-power spectra is given in equa-tion (6). The source of the bias in the observed shear covariance,Cκ

i () − Cκi (), is assumed to be the excursion of dgm = 0.001 in

the mth coefficient of the Chebyshev expansion, others being heldto their fiducial values of zero. We then compare this error to the1σ marginalized error on the parameter pα . For a range of 1 m 30 we find that the biases in the cosmological parameters are en-tirely consistent with statistical degradations shown in Fig. 3. We

also find that the degradations due to higher modes (m 10–20)are progressively suppressed, illustrating again that smooth biasesin redshift are the most important sources of degeneracy with cos-mological parameters, and need the most attention when calibratingthe photometric redshifts.

Finally, let us note that in this analysis we have not attempted tomodel more complicated redshift dependences of bias, the presenceof abrupt degradations in redshift etc. Such an analysis is beyondthe scope of this work, but can be done using similar tools once thedetails about the performance of a given photometric method areknown.

5 R E S U LT S : M U LT I P L I C AT I V E E R RO R

As explained in Section 3.2, we adopt the multiplicative error of theform

Pκi j () = Pκ

i j ()[1 + fi + f j ], (22)

where Pκi j () and Pκ

i j () are estimated and true convergence power,respectively. In other words, the multiplicative error we considercan be thought of as an irreducible but perfectly coherent bias in thecalibration of shear in any given redshift bin.



0 0.001 0.002 0.003 0.004 0.005

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w0

wa

w

Figure 3. Degradation in marginalized errors in M, σ 8 and w = constant, as well as w0 and wa, as a function of our prior knowledge of the redshift biascoefficients gi for the DES, SNAP and LSST. We use N cheb = 30 parameters gi that describe the bias in redshift and give equal prior to each of them, shown onthe x-axis. For the DES and SNAP, knowledge of gi to better than 0.001( f sky/ f sky,fid)−1/2, corresponding to redshift bias of |zp − z s| 0.001( f sky/ f sky,fid)−1/2

for each Chebyshev mode, is desired as it leads to error degradations of about 50 per cent or less. For LSST the requirement is about a factor of 2 stronger.These results are corroborated by computing the bias in cosmological parameters as discussed in the text.

First, note that a tomographic measurement with B redshift binscan determine at most B multiplicative parameters fi (this is trueeither if they are just step-wise excursions as assumed here or co-efficients of Chebyshev polynomials described in the Appendix A).For, if we had more than B multiplicative parameters, at least one binwould have two or more parameters, and there would be an infinitedegeneracy between them. In contrast, the survey can in principledetermine a much larger number of cosmological parameters sincethey enter the observables in a more complicated way. We choose touse exactly N mult = B multiplicative parameters (so N mult = 10 forSNAP and LSST and N mult = 7 for the DES) since we would expectthe shear calibration to be different within each redshift bin.

Fig. 4 shows the degradation in error in measuring three cos-mological parameters as a function of our prior knowledge of themultiplicative factors; we give equal prior to all multiplicative fac-tors. For the DES and SNAP, control of multiplicative error of0.01(f sky/f sky,fid)−1/2 (or 1 per cent in shear for the fiducial sky cov-erages) leads to a 50 per cent increase in the cosmological errorbars, and is therefore about the largest error tolerable. For LSST, therequirement is about a factor of 2 more stringent.

In general, it is interesting to consider whether the weak-lensingsurvey can ‘self-calibrate’, that is, determine both the cosmological

and the nuisance parameters concurrently. This is partly motivatedby the self-calibration of cluster count surveys, where it has beenshown that one can determine the cosmological parameters and theevolution of the mass-observable relation – provided that the lattertakes a relatively simple deterministic form (e.g. Levine, Schulz& White 2002; Hu 2003b; Majumdar & Mohr 2003; Lima & Hu2004). Fig. 4 shows that the fiducial DES and SNAP surveys canself-calibrate with about a 100 per cent degradation in cosmologicalparameter errors (and LSST with about a 150 per cent degradation).While doubling the error in cosmological parameters is a somewhatsteep price to pay, it is very encouraging that all surveys enter aself-calibrating regime where only the higher-order moments of theerror contribute to the total error budget. In Section 7, we showthat the inclusion of BS information can significantly improve theself-calibration regime.

6 R E S U LT S : A D D I T I V E E R RO R

As explained in Section 3.3, we adopt the additive error of the form

Pκadd,i j () = ρbi b j

(

∗

)α

, (23)



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Figure 4. Degradation in marginalized errors in M, σ 8 and w = constant, as well as w0 and wa, as a function of our prior knowledge of the shearmultiplicative factors. We give equal prior to multiplicative factors in all redshift bins, and show results for the DES, SNAP and LSST. For example, existenceof the multiplicative error of 0.01( f sky/ f sky,fid)−1/2 (or 1 per cent in shear for the fiducial sky coverages) in each redshift bin leads to 50 per cent increase inerror bars on M, σ 8 and w for the DES and SNAP, and about a 100 per cent degradation for LSST.

which adds that amount of noise to the convergence power spectra.The coefficient ρ is always 1 for i = j, and its (fixed) value fori = j controls how much additive error leaks into the cross-powerspectra. We weigh the fiducial value of bi by the inverse square ofthe average galaxy size in the ith redshift bin (or, by the square ofthe angular diameter distance to the ith bin).9

Fig. 5 shows the degradations in the equation-of-state w as afunction of the fiducial bi at redshift z = 0.75 where, very roughly,most galaxies are found (recall, the other bi are equal to this valuemodulo order-unity weighting by the square of the angular diameterdistance to their corresponding redshift). The solid line in the figureshows results for our fiducial SNAP survey, assuming no contributionto the cross-power spectra (i.e. ρ = δ i j ). The coefficients bi needto be controlled to ∼2 × 10−5, corresponding to shear varianceof ( + 1)Pκ

add,i j ()/(2π) ∼ 10−4 on scales of ∼10 arcmin ( ∼1000) where most constraining power of weak lensing resides. Theobserved degradation in cosmological parameters is clearly due tothe increased sample variance that the additional power puts on tothe measurement of the PS. When ρ = 1, this sample variance is

9 For bi = 0, the Fisher derivatives with respect to bi are zero and no infor-mation about these parameters can formally be extracted.

confined to a single mode of the shear covariance matrix, so thatthe maximum damage is limited and we observe the self-calibrationplateau in Fig. 5.10

We have tried varying a number of other details.

(i) Using different values of the coefficient ρ for i = j in the range0 ρ < 1;

(ii) changing the fiducial value of the exponent α from 0 to +3or −3;

(iii) adding a 10 per cent prior to the bi (rather than no prior);(iv) using the redshift-independent fiducial values of the coeffi-

cients bi and(v) considering the degradation in the other cosmological

parameters.

Interestingly, we find that the overall requirements are veryweakly dependent on any of the above variations, and the

10 One could in principle also have a worst-case limit with additive errorswhose functional form makes them strongly degenerate with the effect ofvarying the cosmological parameters, where the accuracy in cosmologicalconstraints degrades as soon as the additive power becomes comparable tothe measurement uncertainties in the PS (and not the PS itself, as above).



10-6

10-5

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10-2

fiducial value of bi at z=0.75

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0%90%99%100%

Cross-powercorrelation:

Figure 5. Degradation in the cosmological parameter accuracies as a func-tion of the fiducial value of the additive shear errors bi, assuming no prioron the bi. We show results for SNAP and for several values of the cor-relation coefficients between different bins, ρ, and marginalizing over thespatial power-law exponent α which has fiducial value α = 0. The resultsare insensitive to various details, as discussed in the text, and the only ex-ception is the possibility that the additive effect on the cross-power spectrais negligibly suppressed (i.e. the cross-power correlation is close to 100per cent). We conclude that the mean additive shear will need to be knownto about ∼2 × 10−5, corresponding to shear variance of ∼10−4 on scalesof ∼10 arcmin.

requirements almost always look roughly like those in Fig. 5. Inparticular, the results are essentially insensitive to reasonable priorsof any of the nuisance parameters, since the bi can be determinedinternally to an accuracy much better than bi for all but the smallest(10−7) fiducial values of these parameters. Therefore, the domi-nant effect by far is the fiducial value of bi and the increased samplevariance that it introduces.

The only interesting exception to this insensitivity is allowingthe i = j value of the coefficient ρ to be very close to unity. In theextreme case when ρ = 1 for i = j, the degradation asymptotesto ∼150 per cent even with very large bi since in that case theadditive errors add a huge contribution to all power spectra, and onecan mathematically show that the resulting errors in cosmologicalparameters do not change more than ∼100 per cent irrespectiveof the systematic error. In practice, however, ρ 0.99 for i = jis needed to see an appreciable difference (see the other curvesin Fig. 5), and it is reasonable to believe that the actual errors inthe cross-power spectra will be suppressed by much more than aper cent, resulting in the degradations as shown by the solid curvein Fig. 5.

While our model for the additive errors is admittedly crude, it isvery difficult to parametrize these errors more accurately withoutend-to-end simulations that describe various systematic effects andestimate their contribution to the additive errors (such simulationsare now being planned or carried out by several research groups).Moreover, additive errors cannot be self-calibrated unless we canidentify a functional dependence, which is distinct from the cosmo-logical dependence of the PS, that it must have. Finally, we notethat space-based surveys, such as SNAP, are expected to have amore accurate characterization of the additive error, primarily fromthe absence of atmospheric effects in the characterization of thePSF.

7 S Y S T E M AT I C S W I T H T H E B I S P E C T RU M

We now extend the calculations to the BS of weak gravitational lens-ing. The BS is the Fourier counterpart of the three-point correlationfunction, and it describes the non-Gaussianity of mass distributionin the large-scale structure that is induced by gravitational insta-bility from the primordial Gaussian perturbations that the simplestinflationary models predict. The redshift evolution and configura-tion dependence of the mass BS can be accurately predicted usinga suite of high-resolution N-body simulations (see e.g. Bernardeauet al. 2002, for a comprehensive review). The BS of lensing sheararises from the line-of-sight integration of products of the mass BSand the lensing geometrical factor (see equation 18 in Takada &Jain 2004). Following Jain & Seljak (1997), we roughly estimatehow the lensing PS and BS scale with cosmological parameters byperturbing around the fiducial CDM model:

Pκ ∝ −3.5DE σ 2.9

8 z1.6s |w|0.31, Bκ ∝ −6.1

DE σ 5.98 z1.6

s |w|0.19, (24)

where we have considered multipole mode of l = 1000, equilat-eral triangle configurations in multipole space, redshift of all sourcegalaxies zs = 1 (no tomography), and constant equation-of-state pa-rameter w. We adopt the model described in Takada & Jain (2004) tocompute the lensing BS. Equation (24) illustrates that the PS and theBS depend upon specific combinations of cosmological parameters.For example, the PS (and more generally any two-point statistics ofchoice) depends on DE and σ 8 through the combination −1.2

DE σ8

(or equivalently 0.5M σ 8). Furthermore, the cosmological parameter

dependences are strongly degenerate with source redshift (zs) un-less accurate redshift information is available. Importantly, the PSand BS have substantially different dependences on the parameters,suggesting that combining the two can be a powerful way of break-ing the parameter degeneracies. For example, it is well known thata combination of B/P2, motivated from the hierarchical clusteringansatz, depends mainly on DE, with rather weak dependence onσ 8, so that roughly B/P2 ∝ 0.9

DEσ−0.18 (Bernardeau, Van Waerbeke

& Mellier 1997; Hui 1999; Takada & Jain 2004). As another ex-ample, the BS amplitude increases with the mean source redshift zs

more slowly than P2 because the non-Gaussianity of structure for-mation becomes suppressed at higher redshifts. It is therefore clearthat photometric redshift errors of source galaxies will affect thePS and BS differently. Likewise, it is natural to expect that othersystematics that we have considered affect the PS and the BS in adifferent way.

For the PS plus BS systematic analysis, we consider the redshiftand multiplicative errors but not the additives. For the redshift errorswe adopt the parametrization of redshift bin centroids, now appliedto tomographic bins of both PS and BS. The multiplicative error ismodelled similarly as in Section 3.2: the observed BS with tomo-graphic redshift bins i, j and k, Bi jk(l1, l2, l3), is related to the trueBS Bijk(l 1, l 2, l 3) via

Bi jk(l1, l2, l3) = Bi jk(l1, l2, l3)[1 + fi + f j + fk]. (25)

Based on the considerations above, we study how combiningthe PS with the BS allows us to break parameter degeneracies notonly in cosmological parameter space but also those induced bythe presence of the nuisance systematic parameters. We includeall triangle configurations, and compute the bispectra constructedusing five redshift bins up to l max = 3000. Note that, for a giventriangle configuration, there are 53 tomographic bispectra; the largeamount of time necessary to compute them is the reason why we



Figure 6. Degradation in w0 and wa accuracies expected for SNAP as a function of our prior knowledge of the multiplicative error in shear (left-hand panel)and redshift centroids (right-hand panel). The priors on the multiplicative or redshift parameters are shown on the x-axis. While the PS and BS individuallyenter a self-calibration regime with ∼100 per cent degradation in cosmological parameter errors, combining the two leads to self-calibration with only a20–30 per cent degradation. This improvement in the self-calibration limit is in addition to the already smaller no-systematic error bars with PS plus BS ascompared to either PS or BS alone.

use five tomographic bins instead of the original 7–10.11 Note toothat we have assumed that the PS and BS are uncorrelated andsimply added their Fisher matrices when combining them. Whilethey are strictly uncorrelated in linear theory, non-linear structureformation will introduce the correlation between the two, and the fullinformation content will be smaller than our estimate.12 Correlationbetween the PS and BS has not been accurately estimated to date,and such a project is well beyond the scope of our paper. However,our main emphasis here is not to accurately estimate the resultingcosmological error bars but rather to study the overall effect of thesystematics when different weak-lensing probes are combined. Weexpect the qualitative trends (discussed below) to be unchanged incases when we add information from cross-correlation cosmographyor cluster counts to the PS.

Fig. 6 shows how the multiplicative errors (left-hand panel) andredshift centroid errors (right-hand panel) degrade constraints on w0

and wa expected for SNAP. Each panel shows the degradation in theparameter measurement due to the PS, BS, and the two combined.The most remarkable fact seen in Fig. 6 is that the degradationin the parameter accuracies is smaller with PS and BS combinedas opposed to either one separately. This is because dependences ofBS on cosmological parameters as well as the model systematics arecomplementary to those from PS. Therefore, combining the PS andBS has a very beneficial effect of protecting against the systematicsby more than a factor of 2 than either statistic alone.

However, the drastic improvement in self-calibration does nothold up for parameters that are more accurately measured, such asw = constant. Fig. 7 shows the degradation in the w = constantcase for the multiplicative errors (left-hand panel) and redshift er-rors (right-hand panel), again for SNAP. It is clear that the low self-calibration asymptote seen in Fig. 6 is no longer present, and eventhe cases when PS and BS are combined lead to appreciable degrada-tions. Therefore, w = constant does not benefit from self-calibration

11 For the same reason, the PM curves in Figs 6 and 7 do not exactly matchthose in Figs 2 and 4.12 We thank Martin White for drawing our attention to this issue.

as much as w0 and wa. More generally, self-calibration with PS plusBS works much better when applied to poorly determined combina-tions of cosmological parameters than to the accurately determinedones; we have explicitly checked this by diagonalizing the full Fishermatrix and finding the degradations in all of its eigenvectors.

8 C O M B I N I N G T H E S Y S T E M AT I C E R RO R S

In Table 2, we now show the principal cosmological parameterswith their fiducial values, and their errors for the PS only and PSplus BS cases. In each case we show the error without any system-atics, and the error after adding sample systematic errors. For thesystematics, we assume redshift biases (described by Chebyshevpolynomials) with priors of 0.0005, together with the multiplicativeerrors of 0.005. The errors have been marginalized over the othercosmological parameters, and the systematic-case errors have fur-ther been marginalized over the 30 redshift and 10 multiplicativenuisance parameters with the aforementioned priors. While the sys-tematics used are a guess – and they are likely to remain uncertainwell into the planning phase of each survey – our goal here is to seehow the errors degrade. While the errors in the strongest fiducial sur-veys get degraded the most, we find that strongest surveys remain inthat position even after adding the systematics. For example, fidu-cial error of LSST on w was a factor of 2 better than that of SNAPbefore adding the systematics, and is 50 per cent better after addingthem. Note, however, that we have added equal systematic errorsto all three surveys in this example; in reality, a space-based surveylike SNAP is expected to have a better control of the systematicsthan the ground-based surveys (see e.g. Rhodes et al. 2004b).

In Fig. 8, we show the contours of constant degradation in theequation of state of dark energy w (for the PS case only) with bothredshift and multiplicative errors included. The x-axis and y-axisintercepts of degradation values in this figure correspond to the caseof redshift centroid errors only (as in Fig. 2) and multiplicative errorsonly (as in Fig. 4), respectively, but the rest of the plane allows for thesimultaneous presence of both types of error. Note that the contoursbecome progressively more vertical at larger values of the redshifterror – therefore, the results are weakly affected by the multiplicative



Figure 7. Degradation in w = constant expected for SNAP for the multiplicative errors (left-hand panel) and redshift errors (right-hand panel). Note that thelow self-calibration asymptote seen in Fig. 6 is not seen any more, and even the cases when PS plus BS are combined lead to appreciable degradations. Weconclude that w = constant [or more generally any accurately measured combination of w(z)] does not benefit from self-calibration much, but w0 and wa

individually do as shown in Fig. 6.

Figure 8. Contours of constant degradation in equation of state of dark energy w (for the PS case only) with both redshift and multiplicative errors includedand for the SNAP fiducial survey (left-hand panel) and LSST (right-hand panel). As before, the degradations are unchanged for a different survey area if we letboth errors scale as f −1/2

sky . Note that the contours become progressively more vertical at larger values of the redshift error – therefore, the results are weaklyaffected by the multiplicative errors once the redshift errors become substantial.

errors once the redshift errors become substantial. However, wewould ideally like to be in the regime where the degradation issmaller than ∼50 per cent, and in that case both types of errors (aswell as the additives) need to be controlled to a correspondinglygood accuracy as discussed in Sections 4.1 and 5.

9 C O N C L U S I O N S

We considered three generic types of systematic errors that can affecta weak-lensing survey: measurements of source galaxy redshifts,and multiplicative and additive errors in the measurements of shear.We considered three representative future wide-field surveys (DES,SNAP and LSST) and used weak-lensing tomography with eitherPS alone, or PS and BS combined.

The most important (and difficult) part is to parametrize the sys-tematic errors. We are solely interested in the part of the systemat-ics that has not been corrected for in the data analysis, as that part

can lead to errors in the estimated cosmological parameters. Forthe redshift error we adopt two alternative parametrizations both ofwhich should be useful in calibrating photometric redshift methods.Multiplicative errors in shear measurement are described by one pa-rameter for the error in shear in each given redshift bin. The additiveerrors are most difficult to parametrize, as both their redshift depen-dence and spatial coherence need to be specified; our treatment ofthe additive errors, while robust with respect to various details ofthe model, is only a first step and can be improved upon with furthersimulation of realistic systematics.

In general, higher fiducial accuracy in the cosmological parame-ters leads to more stringent requirements on the systematics. There-fore, LSST typically has the most stringent requirements, followedby SNAP and then the DES (at their fiducial f sky). We note thatthe accurately determined combinations of dark energy parameterstypically are more sensitive to systematics than the poorly deter-mined combinations. At this point one could ask, which is the



important quantity: the individual parameters (say, w0 and wa), ortheir linear combination that is well measured by the survey? Weargue that both are needed: the former to understand the behaviourof dark energy at any given redshift, and the latter to maximize theconstraining power of weak lensing when combining it with othercosmological probes.

For a SNAP-type survey with ∼100 gal arcmin−2, we find thatthe centroids of redshift bins (of width z = 0.3) need to beknown to about 0.003 ( f sky/ f 1000)−1/2 in order not to lead to pa-rameter degradations larger than ∼50 per cent. For the LSST-typesurvey the requirements are about a factor of 2 more stringent,that is, 0.0015 ( f sky/ f 15000)−1/2. These numbers correspond to con-trolling each Chebyshev mode of smooth variations in zp − z s toabout 0.001( f sky/ f 1000)−1/2 and 0.0005 ( f sky/ f 15000)−1/2, respec-tively. These requirements would easily be satisfied by planned sur-veys if the biases were due to residual statistical errors, since thesesurveys will have well over a million galaxies per redshift bin. Butit remains a challenge to control the systematic biases to this level,presumably by using the spectroscopic training sets.

The multiplicative errors need to be determined to about0.01( f sky/ f sky,fid)−1/2 (or 1 per cent of average shear in a given to-mographic redshift bin, for the fiducial sky coverages) for the DESand SNAP. The requirements are about a factor of 2 more stringentfor the LSST. The actual multiplicative error will depend on thegalaxy size and shape, and might have spatial dependence as well.The numbers we quote here refer to the post-correction systematicerror, averaged over all galaxies and directions in the sky.

The additive errors in shear require more detailed modelling thanthe multiplicatives, and in particular, they require specifying its two-point correlation function (and the three-point function if we are toconsider the BS measurements). We constructed a simple modelthat includes the redshift and angular dependence of the additivepower and a coefficient that specifies the effect on the cross-powerspectra relative to that on the auto-spectra. In most cases the additiveerror in each redshift bin needs to be controlled to a few times of10−5, which corresponds to shear variance of ∼10−4 on scales of∼10 arcmin ( ∼ 1000). Note too that the additive error cannotbe self-calibrated unless we can identify a functional dependencethat it must have. Our parametrization of the additive errors is afirst step, and improvements can be made once the sources of theadditive error are studied in more detail both from the data and viaray-tracing simulations for given telescope designs.

While the systematic requirements are not stringent beyond whatone can reasonably hope to achieve with upcoming surveys, perhapsthe most encouraging aspect that we highlighted is the possibility ofself-calibration of a part of the systematics. In this scheme, weak-lensing data are used to concurrently determine both the systematicand the cosmological parameters. The effects of the parametrizedsystematics can then be marginalized out without the need to knowtheir values (but at the expense of increasing the cosmological pa-rameter errors), leaving only the subtler systematic effects that wereeffectively not taken into account with the assumed parametriza-tion. We find that PS measurements can lead to self-calibration with∼100 per cent error degradation in most cases. A promising result isthat combining the PS and BS measurements leads to self-calibrationwith 20–30 per cent degradation, at least for the more poorly con-strained combinations of parameters. Therefore, not only are thefiducial constraints of PS plus BS better than those with PS alone,but also the degradations relative to their fiducial constraints aresmaller in the PS plus BS case, simply because the combined PSplus BS are more effective in breaking the degeneracies betweenthe systematic and cosmological parameters than the PS or the BS

alone. However, we also found that the self-calibration with PS plusBS is not nearly as effective if we consider w = constant (or thebest-determined combination of w0 and wa). More generally, theaccurately measured principal component of w(z) does not benefitfrom self-calibration as much as its poorly measured components.Finally, we only considered the constraints from the PS and BS,without using information from cross-correlation cosmography orcluster counts. Including the latter two methods, which we plan todo in the near future, could further improve the prospects for self-calibration of systematic errors.

AC K N OW L E D G M E N T S

DH is supported by the National Science Foundation (NSF) Astron-omy and Astrophysics Post-doctoral Fellowship under Grant No.0401066. GMB acknowledges support from grant AST-0236702from the NSF, and Department of Energy grant DOE-DE-FG02-95ER40893. We thank Carlos Cunha, Josh Frieman, Mike Jarvis,Lloyd Knox, Marcos Lima, Eric Linder, Erin Sheldon, ZhaomingMa, Joe Mohr, Hiroaki Oyaizu, Fritz Stabenau, Tony Tyson, MartinWhite, and especially Wayne Hu for many useful conversations.

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A P P E N D I X A : C H E B Y S H E V P O LY N O M I A L S

We would like to parametrize the bias between the photometric andspectroscopic redshifts, δz ≡ zp − z s, as a function of zs. This

0 0.5 1 1.5 2 2.5 3

zs

-0.1

-0.05

0

0.05

0.1

0.15

z p - z

s

1

2

7

Figure A1. Select three modes (first, second and seventh) of perturbationto the relation between the photometric and spectroscopic redshift using theChebyshev polynomials out to zmax = 3.

function is expected to be relatively smooth, and one promisingway to parametrize it is to use Chebyshev polynomials. Chebyshevpolynomials of the first kind Ti(x) (i = 0, 1, 2, . . . ) are smoothfunctions, orthonormal in the interval x = [−1, 1], and take valuesfrom −1 to 1. The first two are T 0(x) = 1 and T 1(x) = x .

One can represent the redshift uncertainty in terms of the first MChebyshev polynomials as

δz ≡ zp − zs =M∑

i=1

gi Ti

(zs − zmax/2

zmax/2

), (A1)

where zmax is the maximum extent of the galaxy distribution inredshift. For convenience, the fiducial values for the extra parametersare taken to be gi = 0 (i = 0, 1, 2, . . . , M − 1). Then the derivativeswith respect to the nuisance parameters can be computed via d/dgk =[d/dzp] [dzp/dgk].

Fig. A1 shows the select three modes (first, second and seventh)of perturbation to the relation between the photometric and spectro-scopic redshift.

This paper has been typeset from a TEX/LATEX file prepared by the author.


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